Solve for z square root of z-3=11- square root of z+8

$\sqrt{z - 3} = 11 - \sqrt{z + 8}$

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{z - 3}}^{2} = {(11 - \sqrt{z + 8})}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{z - 3}$ as ${(z - 3)}^{\frac{1}{2}}$ .

${({(z - 3)}^{\frac{1}{2}})}^{2} = {(11 - \sqrt{z + 8})}^{2}$

Simplify the left side.

Simplify ${({(z - 3)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(z - 3)}^{\frac{1}{2}})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(z - 3)}^{\frac{1}{2} \cdot 2} = {(11 - \sqrt{z + 8})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(z - 3)}^{\frac{1}{2} \cdot 2} = {(11 - \sqrt{z + 8})}^{2}$

Rewrite the expression.

${(z - 3)}^{1} = {(11 - \sqrt{z + 8})}^{2}$

Simplify.

$z - 3 = {(11 - \sqrt{z + 8})}^{2}$

Simplify the right side.

Simplify ${(11 - \sqrt{z + 8})}^{2}$ .

Rewrite ${(11 - \sqrt{z + 8})}^{2}$ as $(11 - \sqrt{z + 8}) (11 - \sqrt{z + 8})$ .

$z - 3 = (11 - \sqrt{z + 8}) (11 - \sqrt{z + 8})$

Expand $(11 - \sqrt{z + 8}) (11 - \sqrt{z + 8})$ using the FOIL Method.

Apply the distributive property.

$z - 3 = 11 (11 - \sqrt{z + 8}) - \sqrt{z + 8} (11 - \sqrt{z + 8})$

Apply the distributive property.

$z - 3 = 11 \cdot 11 + 11 (- \sqrt{z + 8}) - \sqrt{z + 8} (11 - \sqrt{z + 8})$

Apply the distributive property.

$z - 3 = 11 \cdot 11 + 11 (- \sqrt{z + 8}) - \sqrt{z + 8} \cdot 11 - \sqrt{z + 8} (- \sqrt{z + 8})$

Simplify and combine like terms.

Simplify each term.

Multiply $11$ by $11$ .

$z - 3 = 121 + 11 (- \sqrt{z + 8}) - \sqrt{z + 8} \cdot 11 - \sqrt{z + 8} (- \sqrt{z + 8})$

Multiply $- 1$ by $11$ .

$z - 3 = 121 - 11 \sqrt{z + 8} - \sqrt{z + 8} \cdot 11 - \sqrt{z + 8} (- \sqrt{z + 8})$

Multiply $11$ by $- 1$ .

$z - 3 = 121 - 11 \sqrt{z + 8} - 11 \sqrt{z + 8} - \sqrt{z + 8} (- \sqrt{z + 8})$

Multiply $- \sqrt{z + 8} (- \sqrt{z + 8})$ .

Multiply $- 1$ by $- 1$ .

$z - 3 = 121 - 11 \sqrt{z + 8} - 11 \sqrt{z + 8} + 1 \sqrt{z + 8} \sqrt{z + 8}$

Multiply $\sqrt{z + 8}$ by $1$ .

$z - 3 = 121 - 11 \sqrt{z + 8} - 11 \sqrt{z + 8} + \sqrt{z + 8} \sqrt{z + 8}$

Raise $\sqrt{z + 8}$ to the power of $1$ .

$z - 3 = 121 - 11 \sqrt{z + 8} - 11 \sqrt{z + 8} + {\sqrt{z + 8}}^{1} \sqrt{z + 8}$

Raise $\sqrt{z + 8}$ to the power of $1$ .

$z - 3 = 121 - 11 \sqrt{z + 8} - 11 \sqrt{z + 8} + {\sqrt{z + 8}}^{1} {\sqrt{z + 8}}^{1}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$z - 3 = 121 - 11 \sqrt{z + 8} - 11 \sqrt{z + 8} + {\sqrt{z + 8}}^{1 + 1}$

Add $1$ and $1$ .

$z - 3 = 121 - 11 \sqrt{z + 8} - 11 \sqrt{z + 8} + {\sqrt{z + 8}}^{2}$

Rewrite ${\sqrt{z + 8}}^{2}$ as $z + 8$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{z + 8}$ as ${(z + 8)}^{\frac{1}{2}}$ .

$z - 3 = 121 - 11 \sqrt{z + 8} - 11 \sqrt{z + 8} + {({(z + 8)}^{\frac{1}{2}})}^{2}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$z - 3 = 121 - 11 \sqrt{z + 8} - 11 \sqrt{z + 8} + {(z + 8)}^{\frac{1}{2} \cdot 2}$

Combine $\frac{1}{2}$ and $2$ .

$z - 3 = 121 - 11 \sqrt{z + 8} - 11 \sqrt{z + 8} + {(z + 8)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$z - 3 = 121 - 11 \sqrt{z + 8} - 11 \sqrt{z + 8} + {(z + 8)}^{\frac{2}{2}}$

Rewrite the expression.

$z - 3 = 121 - 11 \sqrt{z + 8} - 11 \sqrt{z + 8} + {(z + 8)}^{1}$

Simplify.

$z - 3 = 121 - 11 \sqrt{z + 8} - 11 \sqrt{z + 8} + z + 8$

Add $121$ and $8$ .

$z - 3 = 129 - 11 \sqrt{z + 8} - 11 \sqrt{z + 8} + z$

Subtract $11 \sqrt{z + 8}$ from $- 11 \sqrt{z + 8}$ .

$z - 3 = 129 - 22 \sqrt{z + 8} + z$

Solve for $- 22 \sqrt{z + 8}$ .

Rewrite the equation as $129 - 22 \sqrt{z + 8} + z = z - 3$ .

$129 - 22 \sqrt{z + 8} + z = z - 3$

Move all terms not containing $- 22 \sqrt{z + 8}$ to the right side of the equation.

Subtract $129$ from both sides of the equation.

$- 22 \sqrt{z + 8} + z = z - 3 - 129$

Subtract $z$ from both sides of the equation.

$- 22 \sqrt{z + 8} = z - 3 - 129 - z$

Combine the opposite terms in $z - 3 - 129 - z$ .

Subtract $z$ from $z$ .

$- 22 \sqrt{z + 8} = 0 - 3 - 129$

Subtract $3$ from $0$ .

$- 22 \sqrt{z + 8} = - 3 - 129$

Subtract $129$ from $- 3$ .

$- 22 \sqrt{z + 8} = - 132$

To remove the radical on the left side of the equation, square both sides of the equation.

${(- 22 \sqrt{z + 8})}^{2} = {(- 132)}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{z + 8}$ as ${(z + 8)}^{\frac{1}{2}}$ .

${(- 22 {(z + 8)}^{\frac{1}{2}})}^{2} = {(- 132)}^{2}$

Simplify the left side.

Simplify ${(- 22 {(z + 8)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $- 22 {(z + 8)}^{\frac{1}{2}}$ .

${(- 22)}^{2} {({(z + 8)}^{\frac{1}{2}})}^{2} = {(- 132)}^{2}$

Raise $- 22$ to the power of $2$ .

$484 {({(z + 8)}^{\frac{1}{2}})}^{2} = {(- 132)}^{2}$

Multiply the exponents in ${({(z + 8)}^{\frac{1}{2}})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$484 {(z + 8)}^{\frac{1}{2} \cdot 2} = {(- 132)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$484 {(z + 8)}^{\frac{1}{2} \cdot 2} = {(- 132)}^{2}$

Rewrite the expression.

$484 {(z + 8)}^{1} = {(- 132)}^{2}$

Simplify.

$484 (z + 8) = {(- 132)}^{2}$

Apply the distributive property.

$484 z + 484 \cdot 8 = {(- 132)}^{2}$

Multiply $484$ by $8$ .

$484 z + 3872 = {(- 132)}^{2}$

Simplify the right side.

Raise $- 132$ to the power of $2$ .

$484 z + 3872 = 17424$

Solve for $z$ .

Move all terms not containing $z$ to the right side of the equation.

Subtract $3872$ from both sides of the equation.

$484 z = 17424 - 3872$

Subtract $3872$ from $17424$ .

$484 z = 13552$

Divide each term in $484 z = 13552$ by $484$ and simplify.

Divide each term in $484 z = 13552$ by $484$ .

$\frac{484 z}{484} = \frac{13552}{484}$

Simplify the left side.

Cancel the common factor of $484$ .

Cancel the common factor.

$\frac{484 z}{484} = \frac{13552}{484}$

Divide $z$ by $1$ .

$z = \frac{13552}{484}$

Simplify the right side.

Divide $13552$ by $484$ .

$z = 28$

Solve for z square root of z-3=11- square root of z+8

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